On the dangers of using the growth equation on large scales in the Newtonian gauge

TL;DR

This paper investigates the limitations of the standard growth equation in the Newtonian gauge for large-scale cosmological structures, demonstrating significant inaccuracies and proposing a more precise alternative.

Contribution

The authors identify the failure of the growth equation on large scales in the Newtonian gauge and introduce a modified version that improves accuracy while maintaining simplicity.

Findings

The standard growth equation is inaccurate by about 10^4% on large scales in the Newtonian gauge.

A modified growth equation provides significantly better results on large scales.

Results from studies using the standard equation are unreliable on large scales or high redshifts in this gauge.

Abstract

We examine the accuracy of the growth equation $\ddot{\delta} + 2H\dot{\delta} - 4\pi G\rho\delta = 0$, which is ubiquitous in the cosmological literature, in the context of the Newtonian gauge. By comparing the growth predicted by this equation to a numerical solution of the linearized Einstein equations in the $\Lambda$CDM scenario, we show that while this equation is a reliable approximation on small scales ($k\gtrsim $h Mpc$^{-1}$), it can be disastrously inaccurate ($\sim 10^4% $) on larger scales in this gauge. We propose a modified version of the growth equation for the Newtonian gauge, which while preserving the simplicity of the original equation, provides considerably more accurate results. We examine the implications of the failure of the growth equation on a few recent studies, aimed at discriminating general relativity from modified gravity, which use this equation as a starting point. We show that while the results of these studies are valid on small scales, they are not reliable on large scales or high redshifts, if one works in the Newtonian gauge. Finally, we discuss the growth equation in the synchronous gauge and show that the corrections to the Poisson equation are exactly equivalent to the difference between the overdensities in the synchronous and Newtonian gauges.